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We prove the vanishing of the second Hochschild cohomology group      H w 2 ( M , K ( H ) ) H_w^2\,(M,K(H))      , whenever      M ⊂  B ( H ) M \subset B(H)       is a finite countably decomposable von Neumann algebra not containing a non      Γ  \Gamma      -factor or a factor without Cartan subalgebra as a direct summand. Here      H H       is a Hubert space, and      K ( H ) K(H)       the compact operators.

Vanishing of $H\sp 2\sb w(M,K(H))$ for certain finite von Neumann algebras